Procedimiento:
1) Se simplifican cada uno los
radicales dados, factorizando la cantidad subradical hasta dejar
dentro del signo radical un número primo.
2) Se reducen los radicales
semejantes y a continuación los no semejantes.
3) Se simplifican los radicales
reducidos.
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Ejemplos:
a)
Simplificar 2√450 +9√12 -7√48 -3√98
> Factorizando la cantidad
subradical y simplificando:
2√450 =
2√(2)(3²)(5²) = 2(3)(5)√2 = 30√2
9√12 = 9√(2²)(3) =
(9)(2)√3 = 18√3
-7√48 = -7√(3)(4²) =
-7(4)√3 = -28√3
-3√98 = -3√(2)(7²) =
-3(7)√2 = -21√2
> Reduciendo los radicales
semejantes:
→ 30√2 +18√3
-28√3 -21√2
= (30-21)√2 +(18-28)√3
= 9√2 -10√3
Solución.
b) Simplificar √¹/₃ -√⅘
+√⅟₁₂
> Factorizando el denominador de
la cantidad subradical y simplificando:
√¹/₃ = √(1(3)/3(3) =
√3/9 = √(3/3²) = ¹/₃√3
-√⅘ = -√4(5)/5(5) =
-√20/25 = -√2²(5)/5² = -⅖√5
√⅟₁₂ = √¹/(₂²)(₃)
= ¹/₂√¹(³)/(₃)(₃) = ¹/₂√³/(₃²) =¹/₂(¹/₃)√3 = ⅙√3
> Reduciendo los radicales
semejantes y simplificando:
→ ¹/₃√3 -⅖√5
+⅙√3
= (⅓ +⅙)√3 -⅖√5
=
½√3 - ⅖√5 Solución.
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Ejercicio
238.
Simplificar:
1) √45 -√27 -√20
Factorizando la cantidad subradical y simplificando:
√45 = √(5)(3²) = 3√5
-√27 = -√(3)(3²)= -3√3
-√20 = -√(5)(2²) = -2√5
Reduciendo los radicales semejantes y simplificando:
3√5 -2√5 -3√3
= (3-2)√5 + -3√3
= √5 -3√3 Solución.
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2) √175 +√243 -√63 -2√75
Factorizando la cantidad subradical y simplificando:
√175 = √(7)(5²) = 5√7
√243 = √(3)(9²) = 9√3
-√63 = -√(7)(3²) = -3√7
-2√75 = -2√(3)(5²) = 2(-5)√3 = -10√3
Reduciendo los radicales semejantes y simplificando:
5√7 -3√7 +9√3 -10√3
= (5-3)√7 + (9-10)√3
= 2√7 -√3 Solución.
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3) √80 -2√252 +3√405 -3√500
Factorizando la cantidad
subradical y simplificando:
√80 = √(5)(16) = √(5)(4²)
= 4√5
-2√252 = -2√(7)(36) =
-2√(7)(6²) = -2(6)√7 = -12√7
3√405 = 3√(5)(81) =
3√(5)(9²) = 3(9)√5 = 27√5
-3√500 = -3√(5)(100) =
-3√(5)(10²) = -3(10)√5 = -30√5
Reduciendo radicales
semejantes:
→ 4√5 -12√7
+27√5 -30√5
= (4+27-30)√5 -12√7
=
1√5
-12√7
= √5 -12√7 Solución.
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4) 7√450 -4√320 +3√80 -5√800
Factorizando la cantidad subradical y simplificando:
7√450 = 7√(2)(15²) = 7/15)√2 = 105√2
-4√320 = -4√(5)(8²) = -4(8)√5 = -32√5
3√80 = 3√(5)(4²) = 3(4)√5 = 12√5
-5√800 = -5√(2)(20²) = -5(20)√2 = -100√2
Reduciendo radicales semejantes:
105√2 -100√2 -32√5 +12√5
= (105-100)√2 + (-32+12)√5
= 5√2 -20√5 Solución.
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5)
½√12 -⅓√18 + ¾√48 + ⅙√72
Factorizando la cantidad
subradical y simplificando:
½√12 = ½√(3)(4) =
½√(3)(2²) = ½(2)√3 = 1√3 = √3
-⅓√18 = -⅓√(2)(9) =
-⅓√(2)(3²) = -⅓(3)√2 = -1√2
¾√48 = ¾√(3)(16) =
¾√(3)(4²) = ¾(4)√3 = 3√3
⅙√72 = ⅙√(2)(36) =
⅙√(2)(6²) =⅙(6)√2 = 1√2 = √2
Reduciendo radicales
semejantes:
→ √3 -1√2
+3√3 +√2
= (1+3)√3 + (-1+1)√2
= 4√3
Solución.
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6) 3/4√176 -2/3√45 +1/8√320 +1/5√275
Factorizando la cantidad subradical y simplificando:
3/4√176 = 3/4√(11)(4²) = 3/4(4)√(11) = 3√11
-2/3√45 = -2/3√(5)(3²) = -2/3(3)√5 = -2√5
1/8√320 = 1/8√(5)(8²) = 1/8(8)√5 = √5
1/5√275 = 1/5√(11)(5²) = 1/5(5)√11 = √11
Reduciendo radicales semejantes
3√11 +√11 -2√5 +√5
= (3+1)√11 + (-2+1)√5
= 4√11 -√5 Solución.
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7) 1/7√147 -1/5√700 +1/10√28 +1/3√2187
Factorizando la cantidad subradical y simplificando:;
1/7√147 = 1/7√(3)(7²) = 1/7(7)√3 = √3
-1/5√700 = -1/5√(7)(10²) = -1/5(10)√7 = -2√7
1/10√28 = 1/10√(7)(2²) = 1/10(2)√7 = 1/5√7
1/3√2187 = 1/3√(3)(27²) = 1/3(27)√3 = 9√3
Reduciendo radicales semejantes:
√2 + 9√3 -2√7 +1/5√7
= (1+9)√2 + (-2+1/5)√7
= 10√3 -9/5√7 Solución.
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8) √1/3 -√1/2 +√3/4
Factorizando la cantidad subradical y simplificando:
√1/3 = √(1/3)(3/3) = √3/3² = 1/3√3
-√1/2 =- √(1/2)(2/2) = -√2/2² = -1/2√2
√3/4 = √(3/4)(4/4) = √(12)(4² ) = 1/4√(12) = 1/4√(3)(2²) = 1/4(2)√3 = 1/2√3
Reduciendo radicales semejantes:
= 1/3√3 -1/2√2 +1/2√3
= (1/3+1/2√3) + -1/2√2
= 5/6√3 -1/2√2 Solución.
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9)
√⁹/₅ -√⅙ -√⅟₂₀ +√6
Factorizando la cantidad subradical
y simplificando:
√ 9/5 = √ (9/5)(5/5) =
√45/5² = ⅕ √45 = ⅕ √5(3²) = ⅕(3)√5 = ⅗√5
-√⅙ = -√(⅙)(⁶/₆)
= -√6/6² = -⅙ √6
-√⅟₂₀ =
-√(⅟₂₀)(²⁰/₂₀) = -√²⁰/₂₀² = -¹/₂₀√20= -¹/₂₀√5(2²)
= -(¹/₂₀)(2)√5 = - ⅟₁₀√5
√6 = √6
Reduciendo radicales semejantes:
→ ⅗√5 -⅙ √6 – ⅟₁₀√5
+√6
= (⅗ - ⅟₁₀)√5 + (-⅙ +
⁶/₆)√6
= ½√5 + ⁵/₆√6
Solución.
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10) 5/3√3/5 -1/2√3/4 -5√1/15 +3√1/12.
5/3√3/5 = 5/3√(3/5)(5/5) = 5/3√15/5² = (5/3)(1/5)√15 = 1/3√15
-1/2√3/4 = -1/2√(3/4)(4/4) = -1/2√12/4² = -(1/2)(1/4)√12 = -1/8√12 = -1/8√3(2)² = -(1/8)(2)√3 = -1/4√3
-5√1/15 = -5√(1/15)(15/15) = -5√15/15² = (-5)(1/15)√15 = -1/3√15
3√1/12 = 3√(1/12)(12/12) = 3√12/12² = (3)(1/12)√12 = 1/4√12 = 1/4√3(2)² = (1/4)(2)√3 = 1/2√3
1/3√15 -1/3√15 -1/4√3 +1/2√3
= (1/3-1/3)√15 +(-1/4 +1/2)√3
= 0√15 +1/4√3
= 1/4√3 Solución.
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11)
5√128 – ⅓√⅓ -5√98 + √⅟₂₇
Factorizando la cantidad subradical
y simplificando:
5√128 = 5√(2)(64)
=5√2(8²) = 5(8)√2 = 40√2
– ⅓√⅓ = - ⅓√(⅓)(³/₃)
= - ⅓√³/₃² = - ⅓(⅓)√3 = - ⅟₉√3
-5√98 = -5√(2)(7²) =
-5(7)√2 = -35√2
√⅟₂₇ = √(⅟₂₇)(²⁷/₂₇)
= √²⁷/₂₇²= ⅟₂₇√27 = ⅟₂₇√3(3²) = ⅟₂₇(3)√3
= ⅟₉√3
Reduciendo radicales semejantes:
→ 40√2 – ⅟₉√3 -35√2 +⅟₉√3
→ 40√2 – ⅟₉√3 -35√2 +⅟₉√3
= (40-35)√2 + (-⅟₉+⅟₉)√3
= 5√2 + 0√3
=
5√2 Solución.
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12) 2√700 -15√1/45 +4√5/16 -56√1/7
Factorizando la cantidad subradical y simplificando:
2√700 = 2√(7)(10²) = 2(10)√7 = 20√7.
-15√1/45 = -15√(1/45)(45/45) = -15√45/45² = -15(1/45)√45 = -1/3√45 = -1/3√(5)(3²) = -1/3(3)√5 = -√5.
4√5/16 = 4√(5/16)(16/16) = 4√80/16² = 4(1/16)√80 = 1/4√80 = 1/4√(5)(4²) = 1/4(4)√5 = √5.
-56√1/7 = -56√(1/7)(7/7) = -56√7/7² = -56(1/7)√7 = -8√7.
Reduciendo radicales semejantes:
20√7 -8√7 -√5 √5
(20-8)√7 +(-1+1)√5
= 12√7 +0√5 = 12√7 Solución.
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13) √25ax² + √49b - √9ax²
√25ax² = √5²ax² = 5x√a
√49b = √7²b = 7√b
- √9ax² = -√3²ax² = -3x√a
5x√a -3x√a +7√b
= (5x-3x)√a +7√b
= 2x√a +7√b Solución.
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14) 2√m²n - √9m²n + √16mn² - √4mn²
2√m²n = 2m√n
- √9m²n = - √3²m²n = -3m√n
√16mn² = √4²mn² = 4n√m
- √4mn² = -√2²mn² = -2n√m
2m√n -3m√n +4n√m -2n√m
= (2m -3m)√n + (4n -2n)√m
= -m√n +2n√m
= 2n√m -m√n Solución.
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15) a√320x -7√5a²x -(a-4b)√5x
a√320x = a√(5)(8²)x = 8a√5x
-7√5a²x = -7√5a²x = -7a√5x
-(a-4b)√5x = -a+4b√5x
8a√5x -7a√5x -a+4b√5x
= [(8a -7a -a) +4b]√5x
= 0a+4b√5x
= 4b√5x Solución.
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16) √9x-9 +√4x-4 -5√x-1
√9x-9 = √9(x-1) = √9 √x-1 = 3√x-1
√4x-4 = √4(x-1) = √4 √x-1 = 2√x-1
-5√x-1.
= 3√x-1 +2√x-1 -5√x-1
= (3+2-5)√x-1
= 0√x-1 = 0 Solución.
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17) 2√a⁴x+3a⁴y -a²√9x+27y +√25a⁴x+75a⁴y
2√a⁴x+3a⁴y = 2√a⁴(x+3y) = 2√(a²)²x+3y = 2a²√x+3y
-a²√9x+27y = -a²√9(x+3y) = -a²√(3)²(x+3y) = -3a²√x+3y
√25a⁴x+75a⁴y = √25a⁴(x+3y) = √5²(a²)²(x+3y) = 5a²√x+3y
2a²√x+3y -3a²√x+3y +5a²√x+3y
= (2-3+5)a²√x+3y
= 4a²√x+3y. Solución.
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18) 3a√a+1/a² -√4a+4 +(a+1)√1/a+1
3a√a+1/a² = 3a√a+1/a = 3√a+1.
-√4a+4 = -√4(a+1) = -√2²(a+1) = -2√a+1.
(a+1)√1/a+1 = (a+1)√1(a+1)/(a+1)(a+1) = (a+1)√a+1/(a+1)² = (a+1)(1/a+1)√a+1 = 1√a+1
3√a+1 -2√a+1 +1√a+1
= (3-2+1)a+1
= 2a+1 Solución.
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19) (a-b)√a+b/a-b -(a+b)√a-b/a+b + (2a-2b)√1/a-b
(a-b)√a+b /a-b = (a-b)√(a+b)(a-b) /(a-b)(a-b) = (a-b)√a²+b²/(a-b)² = (a-b)(1/a-b)√a²+b² = 1√a²-b²
-(a+b)√a-b/a+b= -(a+b)√(a-b)(a+b)/(a+b)(a+b)= -(a+b)√a²- b²/(a+b)²= -(a+b)(1/a+b)√a²- b²= -1√a²-b²
(2a-2b)√1/a-b = 2(a-b)√(1)(a-b)/(a-b)(a-b) = 2(a-b)√a-b/(a-b)² = 2(a-b)(1/a-b)√a-b = 2√a-b .
1√a²-b² -1√a²-b² + 2√a-b
= (1-1)√a²-b² + 2√a-b
= 0√a²-b² +2√a-b
= 2√a-b Solución
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